Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: The basic transformations

Todays astrometry has reached the micro-arcsecond level in angular measurements of celestial objects. The next generations of astrometric facilities are aiming at the sub-micro-arcsecond scale. Sub-micro-arcsecond astrometry requires a considerable improvement in the theory of light propagation in the curved space-time of the solar system. In particular, it is indispensable to determine light trajectories to the second order of the post-Newtonian scheme, where the monopole and quadrupole structure of some solar system bodies need to be taken into account. In reality, both the light source as well as the observer are located at finite spatial distances from the gravitating body. This fact implies for the need to solve the boundary value problem of light propagation, where the light trajectory is fully determined by the spatial positions of source and observer and its unit direction at past infinity. This problem has been solved in a recent investigation. A practical relativistic model of observational data reduction necessitates the determination of the unit tangent vector along the light trajectory at the spatial position of the observer, which is determined by a sequence of several basic transformations. The determination of this unit tangent vector allows one to calculate the impact of the monopole and quadrupole structure of solar system bodies on light deflection on the sub-micro-arcsecond level, both for stellar light sources as well as for light sources located in the solar system. Numerical values for the magnitude of light deflection caused by the monopole and quadrupole structure of the body are given for grazing light rays at the giant planets. The model GREM is presently used for data reduction of the ESA astrometry mission Gaia. It is shown how the implementation of these basic transformations into GREM would proceed for possible future space astrometry missions.